1
GATE EE 2015 Set 1
MCQ (Single Correct Answer)
+2
-0.6
The signum function is given by $$\mathrm{sgn}\left(\mathrm x\right)=\left\{\begin{array}{l}\frac{\mathrm x}{\left|\mathrm x\right|};\;\mathrm x\neq0\\0\;;\;\;\mathrm x=0\end{array}\right.$$\$ The Fourier series expansion of sgn(cos(t)) has
A
only sine terms with all harmonics.
B
only cosine terms with all harmonics
C
only sine terms with even numbered harmonics.
D
only cosine terms with odd numbered harmonics.
2
GATE EE 2009
MCQ (Single Correct Answer)
+2
-0.6
The Fourier Series coefficients, of a periodic signal $$x\left( t \right),$$ expressed as $$x\left( t \right) = \sum {_{k = - \infty }^\infty {a_k}{e^{j2\pi kt/T}}}$$ are given by
$${a_{ - 2}} = 2 - j1;\,\,{a_{ - 1}} = 0.5 + j0.2;\,\,{a_0} = j2;$$
$${a_1} = 0.5 - j0.2;\,\,{a_2} = 2 + j1;\,\,$$ and
$${a_k} = 0;$$ for $$|k|\,\, > 2.$$

Which of the following is true?

A
$$x(t)$$ has finite energy because only finitely many coefficients are non $$-$$ zero
B
$$x(t)$$ has zero average value because it is periodic
C
the imaginary part of $$x(t)$$ is constant
D
The real part of $$x(t)$$ is even
3
GATE EE 2008
MCQ (Single Correct Answer)
+2
-0.6
Let x(t) be a periodic signal with time period T. Let y(t) = x(t - t0) + x(t + t0) for some t0. The Fourier Series coefficient of y(t) are denoted by bk. If bk=0 for all odd k, then t0 can be equal to
A
T/8
B
T/4
C
T/2
D
2T
4
GATE EE 2007
MCQ (Single Correct Answer)
+2
-0.6
A signal $$x(t)$$ is given by
$$x\left( t \right) = \left\{ {\matrix{ {1, - {\raise0.5ex\hbox{\scriptstyle T} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 4}} < t \le {\raise0.5ex\hbox{\scriptstyle {3T}} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 4}}} \cr { - 1,{\raise0.5ex\hbox{\scriptstyle {3T}} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 4}} < t \le {\raise0.5ex\hbox{\scriptstyle {7T}} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 4}},\,\,\,} \cr { - x\left( {t + T} \right)} \cr } } \right.$$ Which among the following gives the fundamental Fourier term of $$x(t)$$?
A
$${4 \over \pi }\cos \left( {{{\pi T} \over T} - {\pi \over 4}} \right)$$
B
$${\pi \over 4}\cos \left( {{{\pi t} \over {2T}} + {\pi \over 4}} \right)$$
C
$${4 \over \pi }sin\left( {{{\pi t} \over T} - {\pi \over 4}} \right)$$
D
$${\pi \over 4}sin\left( {{{\pi t} \over {2T}} + {\pi \over 4}} \right)$$
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